Three-Dimensional Representation of Traffic Flow

If one numbers vehicles consecutively along a roadway and draws the space-time trajectory of each vehicle on the same x − t graph, then this family of curves can be interpreted as the contours of a three-dimensional surface for which the third dimension is vehicle number n. The intersection of the surface with planes of constant x are the cumulative arrival curves, n vs. t, at the location x. If the surface is smoothed, the orientation of the tangent plane at any point determines the flow q, density k, and car velocity v. All commonly observed properties of traffic flow have simple geometric interpretations in this three-dimensional model.